Kirby Problem 4.58

Definitions and mathematical statement

A Seifert fibered integral homology sphere is a closed oriented three-manifold $Y$ with $H_*(Y;\mathbb Z)\cong H_*(S^3;\mathbb Z)$ and a decomposition into circles locally modeled on a fibered solid torus. The base orbifold is a sphere; write $\Sigma(a_1,\ldots,a_n)$ for its exceptional-fiber multiplicities, taken at least two and pairwise coprime. The oriented Seifert data and the choice of orientation are understood, rather than discarded.

An acyclic filling is a compact connected smooth four-manifold $W$ with $\partial W=Y$ and $\widetilde H_*(W;\mathbb Z)=0$. A contractible filling additionally has trivial homotopy type.

(a) Classify which such $Y$ admit acyclic fillings. In particular, can any example have at least four exceptional fibers? (b) Classify those admitting contractible fillings. Is there a single Seifert homology sphere that has an acyclic filling but no contractible filling?

(c) Can such a $Y$ be a cork boundary in the following sense: there is a contractible smooth $W$ bounded by $Y$ and a self-diffeomorphism $f:Y\to Y$ that does not extend to a diffeomorphism of $W$? Use this broad cork convention; a Stein structure on $W$ is not required.

All fillings here are smooth and compact. Acyclicity does not force their fundamental groups to vanish. Nonextension as a finite group action is also not the same as nonextension of one diffeomorphism, which is the condition in (c).

Short English statement

Which Seifert homology spheres bound acyclic or contractible smooth four-manifolds, and can any be cork boundaries?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2934 (KP-4.58), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 4.58. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.